Optimal. Leaf size=317 \[ \frac {2 \sqrt {-a} \sqrt {\frac {c x^2}{a}+1} \left (a e^2+c d^2\right ) \sqrt {\frac {\sqrt {c} (d+e x)}{\sqrt {-a} e+\sqrt {c} d}} F\left (\sin ^{-1}\left (\frac {\sqrt {1-\frac {\sqrt {c} x}{\sqrt {-a}}}}{\sqrt {2}}\right )|-\frac {2 a e}{\sqrt {-a} \sqrt {c} d-a e}\right )}{3 c^{3/2} \sqrt {a+c x^2} \sqrt {d+e x}}+\frac {2 e \sqrt {a+c x^2} \sqrt {d+e x}}{3 c}-\frac {8 \sqrt {-a} d \sqrt {\frac {c x^2}{a}+1} \sqrt {d+e x} E\left (\sin ^{-1}\left (\frac {\sqrt {1-\frac {\sqrt {c} x}{\sqrt {-a}}}}{\sqrt {2}}\right )|-\frac {2 a e}{\sqrt {-a} \sqrt {c} d-a e}\right )}{3 \sqrt {c} \sqrt {a+c x^2} \sqrt {\frac {\sqrt {c} (d+e x)}{\sqrt {-a} e+\sqrt {c} d}}} \]
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Rubi [A] time = 0.21, antiderivative size = 317, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 5, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.238, Rules used = {743, 844, 719, 424, 419} \[ \frac {2 \sqrt {-a} \sqrt {\frac {c x^2}{a}+1} \left (a e^2+c d^2\right ) \sqrt {\frac {\sqrt {c} (d+e x)}{\sqrt {-a} e+\sqrt {c} d}} F\left (\sin ^{-1}\left (\frac {\sqrt {1-\frac {\sqrt {c} x}{\sqrt {-a}}}}{\sqrt {2}}\right )|-\frac {2 a e}{\sqrt {-a} \sqrt {c} d-a e}\right )}{3 c^{3/2} \sqrt {a+c x^2} \sqrt {d+e x}}+\frac {2 e \sqrt {a+c x^2} \sqrt {d+e x}}{3 c}-\frac {8 \sqrt {-a} d \sqrt {\frac {c x^2}{a}+1} \sqrt {d+e x} E\left (\sin ^{-1}\left (\frac {\sqrt {1-\frac {\sqrt {c} x}{\sqrt {-a}}}}{\sqrt {2}}\right )|-\frac {2 a e}{\sqrt {-a} \sqrt {c} d-a e}\right )}{3 \sqrt {c} \sqrt {a+c x^2} \sqrt {\frac {\sqrt {c} (d+e x)}{\sqrt {-a} e+\sqrt {c} d}}} \]
Antiderivative was successfully verified.
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Rule 419
Rule 424
Rule 719
Rule 743
Rule 844
Rubi steps
\begin {align*} \int \frac {(d+e x)^{3/2}}{\sqrt {a+c x^2}} \, dx &=\frac {2 e \sqrt {d+e x} \sqrt {a+c x^2}}{3 c}+\frac {2 \int \frac {\frac {1}{2} \left (3 c d^2-a e^2\right )+2 c d e x}{\sqrt {d+e x} \sqrt {a+c x^2}} \, dx}{3 c}\\ &=\frac {2 e \sqrt {d+e x} \sqrt {a+c x^2}}{3 c}+\frac {1}{3} (4 d) \int \frac {\sqrt {d+e x}}{\sqrt {a+c x^2}} \, dx-\frac {\left (c d^2+a e^2\right ) \int \frac {1}{\sqrt {d+e x} \sqrt {a+c x^2}} \, dx}{3 c}\\ &=\frac {2 e \sqrt {d+e x} \sqrt {a+c x^2}}{3 c}+\frac {\left (8 a d \sqrt {d+e x} \sqrt {1+\frac {c x^2}{a}}\right ) \operatorname {Subst}\left (\int \frac {\sqrt {1+\frac {2 a \sqrt {c} e x^2}{\sqrt {-a} \left (c d-\frac {a \sqrt {c} e}{\sqrt {-a}}\right )}}}{\sqrt {1-x^2}} \, dx,x,\frac {\sqrt {1-\frac {\sqrt {c} x}{\sqrt {-a}}}}{\sqrt {2}}\right )}{3 \sqrt {-a} \sqrt {c} \sqrt {\frac {c (d+e x)}{c d-\frac {a \sqrt {c} e}{\sqrt {-a}}}} \sqrt {a+c x^2}}-\frac {\left (2 a \left (c d^2+a e^2\right ) \sqrt {\frac {c (d+e x)}{c d-\frac {a \sqrt {c} e}{\sqrt {-a}}}} \sqrt {1+\frac {c x^2}{a}}\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {1-x^2} \sqrt {1+\frac {2 a \sqrt {c} e x^2}{\sqrt {-a} \left (c d-\frac {a \sqrt {c} e}{\sqrt {-a}}\right )}}} \, dx,x,\frac {\sqrt {1-\frac {\sqrt {c} x}{\sqrt {-a}}}}{\sqrt {2}}\right )}{3 \sqrt {-a} c^{3/2} \sqrt {d+e x} \sqrt {a+c x^2}}\\ &=\frac {2 e \sqrt {d+e x} \sqrt {a+c x^2}}{3 c}-\frac {8 \sqrt {-a} d \sqrt {d+e x} \sqrt {1+\frac {c x^2}{a}} E\left (\sin ^{-1}\left (\frac {\sqrt {1-\frac {\sqrt {c} x}{\sqrt {-a}}}}{\sqrt {2}}\right )|-\frac {2 a e}{\sqrt {-a} \sqrt {c} d-a e}\right )}{3 \sqrt {c} \sqrt {\frac {\sqrt {c} (d+e x)}{\sqrt {c} d+\sqrt {-a} e}} \sqrt {a+c x^2}}+\frac {2 \sqrt {-a} \left (c d^2+a e^2\right ) \sqrt {\frac {\sqrt {c} (d+e x)}{\sqrt {c} d+\sqrt {-a} e}} \sqrt {1+\frac {c x^2}{a}} F\left (\sin ^{-1}\left (\frac {\sqrt {1-\frac {\sqrt {c} x}{\sqrt {-a}}}}{\sqrt {2}}\right )|-\frac {2 a e}{\sqrt {-a} \sqrt {c} d-a e}\right )}{3 c^{3/2} \sqrt {d+e x} \sqrt {a+c x^2}}\\ \end {align*}
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Mathematica [C] time = 2.15, size = 445, normalized size = 1.40 \[ \frac {2 \sqrt {d+e x} \left (\frac {i \sqrt {d+e x} \left (4 i \sqrt {a} \sqrt {c} d e-a e^2+3 c d^2\right ) \sqrt {\frac {e \left (x+\frac {i \sqrt {a}}{\sqrt {c}}\right )}{d+e x}} \sqrt {-\frac {-e x+\frac {i \sqrt {a} e}{\sqrt {c}}}{d+e x}} F\left (i \sinh ^{-1}\left (\frac {\sqrt {-d-\frac {i \sqrt {a} e}{\sqrt {c}}}}{\sqrt {d+e x}}\right )|\frac {\sqrt {c} d-i \sqrt {a} e}{\sqrt {c} d+i \sqrt {a} e}\right )}{\sqrt {-d-\frac {i \sqrt {a} e}{\sqrt {c}}}}+\frac {4 d e^2 \left (a+c x^2\right )}{d+e x}+4 i c d \sqrt {d+e x} \sqrt {-d-\frac {i \sqrt {a} e}{\sqrt {c}}} \sqrt {\frac {e \left (x+\frac {i \sqrt {a}}{\sqrt {c}}\right )}{d+e x}} \sqrt {-\frac {-e x+\frac {i \sqrt {a} e}{\sqrt {c}}}{d+e x}} E\left (i \sinh ^{-1}\left (\frac {\sqrt {-d-\frac {i \sqrt {a} e}{\sqrt {c}}}}{\sqrt {d+e x}}\right )|\frac {\sqrt {c} d-i \sqrt {a} e}{\sqrt {c} d+i \sqrt {a} e}\right )+e^2 \left (a+c x^2\right )\right )}{3 c e \sqrt {a+c x^2}} \]
Antiderivative was successfully verified.
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fricas [F] time = 1.03, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {{\left (e x + d\right )}^{\frac {3}{2}}}{\sqrt {c x^{2} + a}}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {{\left (e x + d\right )}^{\frac {3}{2}}}{\sqrt {c x^{2} + a}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.10, size = 978, normalized size = 3.09 \[ \frac {2 \sqrt {e x +d}\, \sqrt {c \,x^{2}+a}\, \left (c^{2} e^{3} x^{3}+c^{2} d \,e^{2} x^{2}-4 \sqrt {-\frac {\left (e x +d \right ) c}{-c d +\sqrt {-a c}\, e}}\, \sqrt {\frac {\left (-c x +\sqrt {-a c}\right ) e}{c d +\sqrt {-a c}\, e}}\, \sqrt {\frac {\left (c x +\sqrt {-a c}\right ) e}{-c d +\sqrt {-a c}\, e}}\, a c d \,e^{2} \EllipticE \left (\sqrt {-\frac {\left (e x +d \right ) c}{-c d +\sqrt {-a c}\, e}}, \sqrt {-\frac {-c d +\sqrt {-a c}\, e}{c d +\sqrt {-a c}\, e}}\right )+3 \sqrt {-\frac {\left (e x +d \right ) c}{-c d +\sqrt {-a c}\, e}}\, \sqrt {\frac {\left (-c x +\sqrt {-a c}\right ) e}{c d +\sqrt {-a c}\, e}}\, \sqrt {\frac {\left (c x +\sqrt {-a c}\right ) e}{-c d +\sqrt {-a c}\, e}}\, a c d \,e^{2} \EllipticF \left (\sqrt {-\frac {\left (e x +d \right ) c}{-c d +\sqrt {-a c}\, e}}, \sqrt {-\frac {-c d +\sqrt {-a c}\, e}{c d +\sqrt {-a c}\, e}}\right )+a c \,e^{3} x -4 \sqrt {-\frac {\left (e x +d \right ) c}{-c d +\sqrt {-a c}\, e}}\, \sqrt {\frac {\left (-c x +\sqrt {-a c}\right ) e}{c d +\sqrt {-a c}\, e}}\, \sqrt {\frac {\left (c x +\sqrt {-a c}\right ) e}{-c d +\sqrt {-a c}\, e}}\, c^{2} d^{3} \EllipticE \left (\sqrt {-\frac {\left (e x +d \right ) c}{-c d +\sqrt {-a c}\, e}}, \sqrt {-\frac {-c d +\sqrt {-a c}\, e}{c d +\sqrt {-a c}\, e}}\right )+3 \sqrt {-\frac {\left (e x +d \right ) c}{-c d +\sqrt {-a c}\, e}}\, \sqrt {\frac {\left (-c x +\sqrt {-a c}\right ) e}{c d +\sqrt {-a c}\, e}}\, \sqrt {\frac {\left (c x +\sqrt {-a c}\right ) e}{-c d +\sqrt {-a c}\, e}}\, c^{2} d^{3} \EllipticF \left (\sqrt {-\frac {\left (e x +d \right ) c}{-c d +\sqrt {-a c}\, e}}, \sqrt {-\frac {-c d +\sqrt {-a c}\, e}{c d +\sqrt {-a c}\, e}}\right )+a c d \,e^{2}+\sqrt {-a c}\, \sqrt {-\frac {\left (e x +d \right ) c}{-c d +\sqrt {-a c}\, e}}\, \sqrt {\frac {\left (-c x +\sqrt {-a c}\right ) e}{c d +\sqrt {-a c}\, e}}\, \sqrt {\frac {\left (c x +\sqrt {-a c}\right ) e}{-c d +\sqrt {-a c}\, e}}\, a \,e^{3} \EllipticF \left (\sqrt {-\frac {\left (e x +d \right ) c}{-c d +\sqrt {-a c}\, e}}, \sqrt {-\frac {-c d +\sqrt {-a c}\, e}{c d +\sqrt {-a c}\, e}}\right )+\sqrt {-a c}\, \sqrt {-\frac {\left (e x +d \right ) c}{-c d +\sqrt {-a c}\, e}}\, \sqrt {\frac {\left (-c x +\sqrt {-a c}\right ) e}{c d +\sqrt {-a c}\, e}}\, \sqrt {\frac {\left (c x +\sqrt {-a c}\right ) e}{-c d +\sqrt {-a c}\, e}}\, c \,d^{2} e \EllipticF \left (\sqrt {-\frac {\left (e x +d \right ) c}{-c d +\sqrt {-a c}\, e}}, \sqrt {-\frac {-c d +\sqrt {-a c}\, e}{c d +\sqrt {-a c}\, e}}\right )\right )}{3 \left (c e \,x^{3}+c d \,x^{2}+a e x +a d \right ) c^{2} e} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {{\left (e x + d\right )}^{\frac {3}{2}}}{\sqrt {c x^{2} + a}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.00 \[ \int \frac {{\left (d+e\,x\right )}^{3/2}}{\sqrt {c\,x^2+a}} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\left (d + e x\right )^{\frac {3}{2}}}{\sqrt {a + c x^{2}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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